SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus

TL;DR

This paper extends the Finite Element Exterior Calculus framework to Summation-by-Parts finite difference methods, enabling divergence- and curl-free discretizations without known basis functions, through a novel analytic relationship.

Contribution

It introduces a new approach to construct compatible operators in SBP-FD methods inspired by FEEC, despite the lack of basis functions.

Findings

Achieves divergence- and curl-free discretizations in SBP-FD methods.

Develops a method to construct compatible operators using integral and nodal degrees of freedom.

Demonstrates the applicability of FEEC principles to SBP finite difference schemes.

Abstract

We demonstrate that we can carry over the strategy of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) methods to achieve divergence- and curl-free discretizations. This is not obvious at first sight, as for SBP-FD no basis functions are known, but only values and derivatives at points. The key is a remarkable analytic relationship that enables us to construct compatible operators using integral and nodal degrees of freedom. Pre-existing SBP-FD matrix operators can then be used to obtain nodal values from the integral degrees of freedom to derive a scheme with the desired properties.